Stochastic limit-cycle oscillations of a nonlinear system under random perturbations

The paper’s Theorem 3.4 rigorously establishes the local entropy balance on a stable limit cycle with three equivalent expressions dS_l/dt = ∇·γ = − d/dt ln ω = d/dt ln ||γ|| + (1/2) d/dt ln v, using (i) the WKB decomposition b = −D∇ϕ + γ and ∇ϕ ≡ 0 on Γ; (ii) the stationary transport relation ∇·(ωγ) = 0 on Γ implying − d ln ω/dt = ∇·γ; (iii) the periodic Lyapunov/Riccati structure for [Σ*(t)]^{-1} and a careful treatment of the zero tangential eigenvalue to show d ln λ_1/dt = −2 d ln ||γ||, yielding the third expression; and (iv) the constancy of √v ||ωγ|| on Γ (Theorem 3.3), which is consistent with the chain of equalities (see Eqs. (3.10)–(3.12), (3.41)–(3.43), (3.45)–(3.55) in the PDF). The candidate solution erroneously identifies the local entropy rate with (1/2) d/dt ln v by restricting to the normal bundle and thus misses the indispensable tangential-speed contribution d/dt ln ||γ||; it subsequently claims that dS_l/dt = ∇·γ only under constant speed or after redefining a “tube entropy,” which contradicts the paper’s general result (cf. Eq. (3.16), (3.50)–(3.51), and (3.55) in the paper).